01/30/2024 – Audriana Houtz: Connectivity of the Cut-System Complex
abstract: We will start with an introduction to cut-systems on a handlebody. Then prove that the cut-system complex where cut-systems are related by simple moves is connected and if time allows we will prove that it is simply connected.
02/06/2024 – Connor Malin: Transforms in calculus and category theory
abstract: We recall classical Fourier analysis and its applications before describing the general setting of harmonic analysis and Pontryagin duality. We then describe a categorical analog of the Fourier transform and compute it in several cases.
02/13/2024 – Andy Yang: Logics for Modern Language Processing
abstract: Transformers have gained prominence in natural language processing (NLP), both in direct applications like machine translation and in pretrained models like GPT. Today, we sketch out some connections between transformers and first-order logic, temporal logic, as well as algebraic automata theory.
02/20/2024 – Katherine Novey: Intro to Groups of Homotopy Spheres
abstract: In this talk, I will discuss the first few sections of “Groups of Homotopy Spheres” by John Milnor and Michel Kervaire. I will begin by introducing $\Theta_n$, the $n$th group of h-cobordism classes of homotopy spheres, and how it can be calculated in terms of a particular subgroup, the stable homotopy groups of spheres, and the $J$-homomorphism. I will then discuss how to show that this subgroup is trivial in the case $n=2k$, focusing on the computation for even values of $k$.
02/27/2024 – Xiyan Zheng: Prym representations and twisted cohomology of the mapping class group with level structures.
abstract: For a finite cyclic cover S_p of a non-closed surface S of finite type, the whole mapping class group of S does not lift to the cover S_p. However, there is a finite-index subgroup which does lift to the cover and act on the first rational cohomology group of S_p, and this is called a Prym representation. We computed the twisted cohomology of this subgroup in coefficients r-tensor powers of H^1(S_p;Q), which turns out to be unstable with respect to the genus of the surface when r＞2, contrasting to the stability of cohomology of the whole mapping class groups. As a corollary, we also know the Prym representations of finite abelian covers are locally rigid.
Arxiv: http://arxiv.org/abs/2401.13869
03/04/2024 – Hari Rau-Murthy: An HKR theorem for factorization homology, and a related project in character theory
abstract: This talk will primarily be on a generalization of the so-called “HKR theorem”. The classical HKR theorem concerns two different ways of generalizing the notion of differential forms on a ring; the theorem asserts they agree. I will give a generalization of this theorem for ring spectra which will allow us to recalculate the factorization homology/higher THH of rational KU.
03/05/2024 – Cory Gillette: An Introduction to (Co)End Calculus and the Yoneda Lemma
abstract: After a friendly but brief discussion of basic concepts in category theory, we will state the Yoneda lemma and meditate on its proof. Then we will introduce more suggestive notation in terms of (co)ends, and restate the lemma. Finally, we will sketch a handful of applications.
03/27/2024 – Juan Ramirez: Rees Algebras and Rational Plane Curves
abstract: Consider the polynomial ring R=k[x,y] in two variables over a field k and I an ideal presented by a class of 3 x 2 matrices (Hilbert-Burch) with column degrees d_1 \leq d_2. In this talk we will say a few words on how one can translate information about the defining equations of the Rees Algebra of I and Rational Plane Curves of degree d_1+d_2=d. In particular, how one can translate between the configuration of the singularities of a Rational Plane Curve C and the bi-degrees of the defining equations of the Rees Algebra R(I).
04/09/2024 – Jason Mitrovich: The Plateau Problem and Brouwer’s Fixed Point Theorem
abstract: This seminar will introduce the Plateau problem, which is physically motivated by soap films and is named after the physicist Joseph Plateau. First, we will state the general Plateau problem before narrowing down to two simpler cases. Then we will sketch out how standard analysis techniques, as well as fixed points, can be used to solve the simpler cases. Afterwards, we will return to the general setting; however, to make progress in this setting, geometric measure theory (GMT) was developed to solve this problem and ultimately was used to resolve the Plateau problem. To close, we will discuss some results on the regularity of such surfaces as well as the higher mapping problem.
04/16/2024 – Khoi Nguyen: Urbano Theorem and Minimal Surfaces in S^3
abstract: Studying minimal surfaces in different Riemannian manifolds is a recurring theme in the field of geometric analysis. In this talk, I will state a result characterizing closed, orientable minimal surfaces of the 3-sphere. I will start by reviewing the basic theory of minimal surfaces and explaining the relevant terms in the understanding of Urbano theorem. After that, I will state and give a proof of the theorem, which highlights the role of the Clifford torus in studying such minimal surfaces.
04/23/2024 – Postdoc Panel
abstract: This session will focus on how to apply for research postdocs in math. First, we’ll quickly go over the requirements for postdoc applications along with the typical timeline of what happens when, and then we’ll open up a Q&A with current ND students who are about to start research postdocs next year. Students from all years are welcome but it will be particularly relevant to students about to enter the final year of the Ph.D.
04/30/2024 – Lorenzo Riva: The Why, What, and How of higher categories
abstract: One fundamental technique in mathematics is the process that we all know as “abstraction” but that in some cases goes by the name “categorification”: we take some theorem/calculation/construction X and we realize it as the shadow of a more complicated thing Y in a way that makes the concept X more natural, or its construction more elucidating, or its applications more useful in other areas of math. In this talk we will use the homotopy theory of topological spaces and categorify it to obtain a flavor of categorical objects called higher categories, an extension of ordinary categories. This will be an expository talk and no previous knowledge of homotopy theory (besides some basic topology and algebra) will be assumed.
05/03/2024 – Jiayi Shen: Liftableness of torsion elements in GL_n(Z)
abstract: Consider the representation $\text{Out}(F_n)\rightarrow\text{GL}_n(\mathbb{Z})$. We would like to answer the question whether all torsion elements in $\text{GL}_n(\mathbb{Z})$ can be lifted to torsion elements in $\text{Out}(F_n)$. We use a geometric approach towards torsion elements in $\text{Out}(F_n)$ and use algebraic number theory to study torsion elements in $\text{GL}_n(\mathbb{Z})$. For a matrix with prime order $p$ where $p\leq 19$, it is always liftable. For $n\geq 22$, there always exists an order $23$ matrix that can not be lifted to $\text{Out}(F_n)$.